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The cold box with heat exchanger platefin (PFHE) has been applied in various applications, including air separation units (ASU). Therefore, this cryogenic industry has undergone a lot of development in recent years. Cryogenic technologies are utilized in many industrial procedures where they aid in heat recovery and reduce energy consumption. The multistream platefin heat exchanger (MSPFHE) is a substantial part of the air separation plant design. In this study, the energy contained in streams resulting from the distillation tower is used to cool the air entering this process to below freezing temperatures (74 degrees Celsius). In the beginning, to find a suitable model and design, analytical solutions were used to find the heat duty and the optimal heat transfer area, including thermodynamic properties calculations, fin geometry dimensions, heat transfer calculations, and pressure drop calculations. The heat exchanger's design has also been evaluated analytically by calculating the fin efficiency and overall efficiency. A MATLAB code was written to achieve speed, accuracy, and consideration of all possible values. Input data from one of the studies in the literature was used as input for a case study. Finally, the rating of the design of the heat exchanger was done by one of the wellknown software programs (EDR), and the results of this work were compared to those of previously published studies and they were found to be good and compatible.
thermodynamic properties, multistream platefin heat exchanger (MSPFHE), analytical solutions, aspen EDR, MATLAB
Due to its small design, high level of efficiency (greater than 90%), and capacity to handle numerous gas streams, plate fin heat exchangers (PFHE) are used in the industry, particularly for cryogenic processes [1]. One of the main applications is air separation units (ASU), which typically operate under the following conditions: a temperature difference between the cold and warm ends of between 100K and 300K; a pressure drop of between a few tens and hundreds of Pascal; and a temperature difference between the cold and warm streams of between a few Kelvin degrees. A variety of process and mechanical design elements have been examined, and innovative banking arrangements, exchanger surfaces, and flow compensation procedures have been offered in order to increase efficiency in a number of sources that deal with developing platefin heat exchangers [2]. In a society that values the environment and saving energy, MSPFHEs are becoming more and more popular. On the other hand, platefin heat exchangers can be created and constructed for a wide range of applications and flow configurations. The MSPFHE countercurrent design, which has been utilized to handle cryogenic gases in air separation plants, is the primary subject of this work [3].
A classification system based on construction, operation, flow, and heat transfer was devised due to the enormous variety of heat exchanger combinations. The demand for improved performance parameters while maintaining the heat exchanger volume and weight relatively low gave rise to a new generation of heat exchangers known as compressed heat exchangers. Compact heat exchangers (CHE) (Extended Surface Exchangers of Heat) are one of the most important features of many cryogenic parts because they have a high heat transfer surface to volume ratio [4]. Compact heat exchangers (CHE) are heat exchangers that have more than 700 m^{2}/m^{3} of heat transfer surface area per unit volume (CHE). Compact heat exchangers are frequently used in gastoliquid (liquidtogas) and gastogas heat exchange, where the large surface area also aids in boosting the low heat transfer coefficient associated with the gas stream. To lower the amount of energy used in these operations, a variety of effective and small pieces of equipment have been developed, with multistream platefin heat exchangers being one of the best options. As a result, this paper will address the design of this kind of exchanger.
Conventional heat exchangers, such shell and tube heat exchangers, are defined as heat exchangers with a prime surface that normally has an efficacy of 60% or less and a surface area density of the heat transfer that is typically 700 m^{2}/m^{3} [5]. Because the mass and volume of the box are constrained in particular applications and the increased effectiveness of the exchanger (up to about 98%) is crucial, a much smaller surface area has been assigned. Additionally, the coefficient of heat transmission in heat exchangers with gases or other fluids is quite low on one or both sides of the fluid, necessitating a high heat transfer surface area. The addition of extended surface (i.e., fins) and usage of the fins with a fin density that is as high as feasible on one fluid side or both of them, depending on design requirements, is a fairly typical way for enhancing exchanger compactness and surface area [6]. An "extended surface exchanger" has been the name given to the resulting exchanger. By using thingauge material and properly sizing the core, the flow area is increased. In comparison to the unfinned surfaces, the extended surfaces' coefficient of heat transmission may be lower or higher. For example, internal tube fins enhance the surface area of the tube side, but interrupted (louvre, strip, and so on) fins show an increase in area as well as an increase in coefficient of heat transfer. Nevertheless, depending on the fin spacing, they can result in a modest reduction in the coefficient of heat transmission. In general, a decrease in the fins' coefficient of heat transmission follows an increase in fin density. The coefficient of heat transfer may rise by 24 times that of the comparable plain surface of the fin due to flow interruptions (such as those caused by louvred fins, offset strip fins, and so on) [6].
A compact heat exchanger (CHE), of which PFHE is one form, must have a surfacetovolume ratio greater than 700 m2/m3. Several platefin heat exchangers are 23 times more compact than traditional heat exchangers. They are able to transfer heat at such a rapid rate thanks to their special design. Figure 1 depicts a platefin heat exchanger with parallel plates (also known as separating sheets) joined by fins with different shapes. These fin tubes provide the flow channel for different fluid streams. We utilize thicker sidebars to close the flow path. Fins serve a special purpose in these heat exchangers. They support structural stiffness in addition to assisting with heat transmission.
Figure 1. A stack of fins is stacked between the separating sheets [7]
A platefin heat exchanger's fins aid in heat transfer in a number of different ways. The heat is initially transferred from a particular plate to the fluid stream when they are used as secondary surfaces (primary surface). When two neighbouring plates are joined by fins, a parallel path of heat transmission by conduction develops. Last but not least, common fin features may disturb the boundary layer and raise the local convective coefficient of heat transmission. The most typical fin sheets used in platefin heat exchangers are depicted in Figure 2 [8]. The influence of operating parameters on platefin heat exchangers (PFHE) was examined by Patil and Rathod [9]. The offset strip fins (OSF) with rectangular cross sections of the platefin heat exchanger are examined. MATLAB is used to create a steadystate model for the core dimensions of PFHE with a crossflow configuration.
Tan et al. [10] studied the effects of properties on heat exchanger sizing design led to the identification of the crucial characteristics that had to be given top priority when creating a property model. Besides, the authors reported that even though viscosity has less significant impact compared to other properties, the larger deviation range of current viscosity models may lead to higher uncertainties in volume design and annual capital cost of heat exchanger. A technique was created by GarcíaCastillo & PicónNúñez [11] for figuring out the surface geometry that will produce the necessary pressure drop and heat transfer coefficient to meet the design criteria. The application of this methodology to a case study showed that a shell and tube heat exchanger of 227.4 m^{2}, with the appropriate fin density using offset stripfins, could be replaced by a plate and fin exchanger with any combination of height, width, and length in the ranges of 0–0.58 m, 0–0.58 m, and 0–3.59 m. This heat exchanger can be twostream or multistream. Figure 3 shows the main structure of a multistream fin plate heat exchanger (MSPFHE). It has a stub pipe, a header tank, a distributed fin, a heat transfer fin, a partition plate, a sidebar, and a cover plate [12].
Figure 2. Various different forms of fins
Figure 3. Fourstream platefin heat exchanger [12]
The air is drawn into air separation units (ASU), which are found near factories. Filtered and cryogenically distilled air is used. Heat exchangers that operate at cryogenic temperatures are the most important components of an air separation plant. Heat exchangers with an efficiency of <90% could result in the failure of the plant, leading to the use of platefin heat exchangers with a large value of surface area per unit volume and low fluid pressure drop [13].
Although previous studies have discussed the design of the heat exchanger, there is a need for many more studies in an effort to obtain the optimal design. Therefore, the present study focuses on the platefin heat exchanger (PFHE) as a critical component in the design of air separation units. The energy contained in streams emanating from the distillation tower in the air separation unit (ASU) has been utilized to cool the air entering this operation in order to achieve reduced energy consumption. A multistream platefin heat exchanger (MSPFHE) has been designed based on one of the models. Firstly, MATLAB code using the technique of a certain model has been prepared, and the code obtained in some practical cases has been applied. This code has been used for sizing platefin heat exchangers in a number of cases with satisfactory results, which included thermodynamic property calculations, heat transfer calculations, pressure drop calculations, fintype selection, and final heat exchanger dimensions. Here also, inconstant properties have been adopted as a fundamental element of obtaining the high effectiveness that this type requires. The study goal is to model and design a multistream platefin heat exchanger (MSPFHE) that can be applied practically without restrictions in the air separation unit cold box. Finally, the results have been validated and rated using the Aspen EDR software, based on the program's capacity to replicate the same environmental conditions that nature creates. An analytical solution and code were used to determine the heat exchanger's primary dimensions as well as the rest of the work's specifics. The efficiency was computed using the current findings and was discovered to be higher than 98%, which is required for this kind of application, as well as a confirmation of the findings by comparing them to prior literature. The current paper's outline is included, with sections 2—a mathematical model; 3—a design methodology; 4—design calculations and estimations; 5—results and discussion; and 6—conclusion.
2.1 Assumptions
All of the fluids in the heat exchanger, which have been depicted in Figure 4, are superheated, i.e., no change in the phase can occur. Also, the working hypotheses below make it less likely that heat will be transferred:
(1) The cold side (N_{b}) fin layer number has been assumed to be one higher than the hot side (N_{a}).
(2) Steadystate conditions.
(3) The study assume that the heat transfer coefficient, thermal capacities, and distribution area are all the same and stay the same.
(4) The thermal walls' resistance is ignored.
(5) Fouling is not noticeable for a heat exchanger of gastogas; therefore, it is ignored.
(6) Axial heat conduction is negligible.
Figure 4. Schematics of a section of the heat exchanger unit [14]
2.2 Thermal modeling
Figure 5 shows a diagram of a typical parallel and counterflow platefin heat exchanger with an offset strip fin, and Table 1 shows how to do thermal modelling of a platefin heat exchanger.
Figure 5. Schematic diagram of parallel and counter flow plate fin heat exchanger with rectangular offset strip fin [15]
Table 1. Formulation related to thermal modeling [15]
NO 
Equations 
Remarks 
1 
$\varepsilon=1\exp \left\{\left(\frac{1}{C^*}\right) N T U^{0.22} *\left(\exp \left\{C^*(N T U)^{0.78}1\right)\right\}\right.$ 
Effectiveness of counter flow HE 
2 
$N T U_{\max }=\frac{U A_{\text {tot }}}{C_{\min }}$ 
Number of transfer units 
3 
$C^*=\frac{C_{\min }}{C_{\max }}$ 
Total heat capacity ratio 
4 
$U=\frac{1}{\frac{1}{\left(h \eta_s\right)}+\frac{1}{\frac{A_\,{t o t, h}}{A_\,{t o t, c}}\,\left(h \eta_s\right)\,_h}}$ 
Overall heat transfer coefficient 
5 
$\beta=\frac{A_{c e l l}}{V_{c e l l}}=\frac{2\left(bt_f\right) x+2\left(ct_f\right) x+x\left(bt_f\right) t_f+c t_f}{b c x}$ 
Surface area density 
6 
$\dot{Q}=C_h\left(T_{h, 1}T_{h, 2}\right)$ 
Rate of heat transfer in hot side 
7 
$\dot{Q}=C_c\left(T_{c, 2}T_{c, 1}\right)$ 
Rate of heat transfer in cold side 
8 
$V_p=L_c L_h b N$ 
Volume between all two plates 
9 
$N=\frac{L_nb_c+2 t_w}{b_h+b_c+2 t_w}$ 
Total passages 
10 
$\eta_s=1\frac{A_f}{A_{\text {cell }}}\left(1\eta_f\right)$ 
Overall surface efficiency 
Figure 6. Hot and cold streams in recuperative heat exchangers [14]
Heat exchange occurs between cold products and hot air, as shown in Figure 6. In the case study (data shown in Table 2), multistream heat exchangers have been used to cool the incoming warm air stream with the cold gases produced by distillation towers in the air separation unit, reducing the need for external refrigeration.
Table 2. Design data provided by Boehme et al. [14]
Stream 
Air 
WN_{2} 
GO_{2} 
LPN_{2} 
HPN_{2} 
Mass Flow Rate(kg/h) 
17,258.6 
10,958.3 
3432.8 
1798.6 
1001.1 
Cold End Pressure (kPa) 
678.3 
121.6 
135.2 
128.4 
544.5 
Cold End Temp. (K) 
195.7 
193.15 
193.15 
193.15 
193.15 
Hot End Temp. (K) 
311.15 
308.15 
308.15 
308.15 
308.15 
2.3 Dimensioning of the heat exchanger
The dimensions of the HEX need to be decided so that the heat exchanger may operate soundly in its operating pressure ambit. Which is why the maximal limits of the width, height, and length should not be exceeded by 1.2 m, 1.2 m, and 6.2 m [16]. The length has to be in the flow direction. The width value has been assumed within the limit that has been characterised before, and then other dimensions are estimated according to the width.
2.4 Fluid thermo physical properties
The important step in the design process of HEX represents a definition of the physical characteristics of the materials [17]. Regarding solid materials, just the equations of conduction are solved for steadystate problems, so the value of thermal conductivity is wanted as an average quality. However, fluid characteristics like thermal conductivity, density, and viscosity are needed for design calculations. There are two assumptions, the first one depends on the independence of temperature and the other one depends on the dependence of temperature. In the present study, the thermophysical characteristics of the fluids have been assumed to be temperatureindependent. By using the ASPEN HYSES simulation program, it can get the properties in Tables 3, 4, 5, and 6.
Table 3. c_{p} of five streams calculated by ASPEN HYSES
Stream 
c_{p} (J/kg.C) 

In 
Out 
Average 

LPN2 
1026.66 
1044.16 
1035.415 
Air 
1004.04 
1003.69 
1003.87 
WN2 
1026.31 
1044.07 
1035.19 
GO2 
883.35 
916.66 
900.01 
HPN2 
1048.53 
1051.69 
1050.11 
Table 4. Density of five streams calculated by ASPEN HYSES
Stream 
Density (kg/m^{3}) 

In 
Out 
Average 

LPN2 
2.2487 
1.2779 
1.7633 
Air 
7.5737 
12.1993 
9.8865 
WN2 
2.1292 
1.2230 
1.6761 
GO2 
2.7087 
1.6237 
2.1662 
HPN2 
9.6606 
5.9265 
7.7935 
Table 5. Viscosity of five streams calculated by ASPEN HYSES
Stream 
Viscosity (Pa.s) E05 

In 
Out 
Average 

LPN2 
1.2792 
1.8800 
1.5800 
Air 
1.9550 
1.3500 
1.6500 
WN2 
1.2791 
1.8800 
1.5800 
GO2 
1.4290 
2.1500 
1.7900 
HPN2 
1.2903 
1.8800 
1.5800 
Table 6. Thermal Conductivity of five streams calculated by ASPEN HYSES
Stream 
Thermal Conductivity (W/m.k) 

In 
Out 
Average 

LPN2 
1.79E02 
2.65E02 
2.22E02 
Air 
2.59E02 
2.59E02 
2.59E02 
WN2 
1.79E02 
0.026511 
2.22E02 
GO2 
1.76E02 
2.71E02 
2.24E02 
HPN2 
1.83E02 
2.67E02 
2.25E02 
2.5 Analytical solution
3.1 Estimation of heat duty
T_{Air in}=311.15K, T_{Air out}=195.7K
$C_{a i r}=\dot{m}_{A i r} C p_{A i r}=4812.608 \mathrm{~W} /{ }^{\circ} \mathrm{C}$
∆T_{Air}=311.15195.7=115.45 K=115.45℃
Q_{Air}=C_{Air} ∆T_{Air}=555.616 kW
Based on assume, the problem steady state: $Q_{A i r}=Q_{L P N_2}+Q_{W N_2}+Q_{G O_2}+Q_{H P N_2}$.
·Heat lost by Air=Heat gained by 4 gases
·Heat duty=555.616 Kw
·In the same way, it calculates the heat loads of other streams as shown in Table 7.
Table 7. Calculation the heat loads of other streams
Stream 
Air 
WN_{2} 
GO_{2} 
LPN_{2} 
HPN_{2} 
Heat load (kW) 
555.616 
362.38 
98.50 
59.50 
33.58 
3.2 Estimation of overall coefficient heat transfer
By use diagram in Figure 7 below, U=85 W/m^{2}℃.
Figure 7. Diagram to calculate the overall heat transfer coefficient [18]
3.3 Estimation of log mean temperature difference
Figure 8. Diagram to calculate the correction factor [18]
Based on Figure 8, the estimation of the Log Mean Temperature Difference is shown below:
·For hot stream (Air): T_{h in}=311.15 K, and T_{h out}=195.7 K
·For cold stream, (There are four cold streams):
$\begin{gathered}T_{\text {c in }}=T_{L P N_2}=T_{W N_2}=T_{G O_2}=T_{H P N_2}=193.15 \mathrm{~K} \\ T_{\text {cout }}=T_{L P N_2}=T_{W N_2}=T_{G O_2}=T_{H P N_2}=308.15 \mathrm{~K} \\ \Delta T_{\text {lm }}=\frac{\left(T_{\text {hin }}T_{\text {cout }}\right)\left(T_{\text {hout }}T_{\text {cin }}\right)}{\ln \left[\frac{\left(T_{h \text { hin }}T_{\text {cout }}\right)}{\left(T_{\text {hout }}T_{\text {cin }}\right)}\right]}=2.769{ }^{\circ} \mathrm{C} \\ N T U=1.06\end{gathered}$
·By use diagram in Figure 8 below, F_{t}=0.97.
·Estimate (∆T_{m}), ∆T_{m}=∆T_{lm}*F_{t}=2.67℃.
3.4 Estimation of area of heat transfer
The optimum required area for heat transfer:
$A_{\text {air }}=\frac{Q}{U * \Delta T_m}=2447.648 \mathrm{~m}^2$
Then, In the same way, we calculate the area of heat transfer of other streams as shown in Table 8:
Table 8. Calculation of the area of heat transfer of other streams
Stream 
Air 
WN_{2} 
GO_{2} 
LPN_{2} 
HPN_{2} 
Area (m^{2}) 
2447.648 
1596.3 
433.9 
262.11 
147.9 
3.5 Estimations of geometry of the heat exchanger
The size of a brazed aluminum platefin heat exchanger shall be specified by width W, stacking height H, and length L of the rectangular block. By trial and error method, the authors calculated the suitable length (L_{2}), L_{2}=4.03 m and height (L_{3}): L_{3}=1.146 m. Figure 9 display fin geometries of the case study which was used in the present work.
Figure 9. Offset strip fin geometry under consideration [19]
The allowed ranges of the design variables are as follows:
Accordingly, it can determine the design information of the geometry of the heat exchanger that is shown below in Table 9 as follows:
Table 9. Design of geometry details
Particulars 
Hot side 
Cold side 
Thickness of fins, (δ) (mm) 
0.25 
0.25 
Thickness of plate (δ_{w}) (mm) 
2 
2 
Height of fin (b) (mm) 
6 
5 
Length of fin, (λ) (mm) 
8 
7 
Density of fin (N_{f}) 
402 
672 
Width of fin, (p_{f}) (mm) 
1.8 
1.488 
3.5.1 Calculations of hot side
·The number of passages for the hot fluid side be: $\mathrm{N}_{\mathrm{p}}=\frac{\mathrm{L}_3\mathrm{b}_22 \delta_{\mathrm{W}}}{\mathrm{b}_1+\mathrm{b}_2+2 \delta_{\mathrm{W}}}=63$
·The total number of fins for hot fluid are calculated by: $\mathrm{n}_{\mathrm{f} 1}=\frac{\mathrm{L}_1}{\mathrm{p}_{\mathrm{f}}} * \mathrm{~N}_{\mathrm{p}}=42 * 10^3$
·The total primary area (A_{p}) for hot fluid (air) calculated by:
$\begin{aligned} \mathrm{A}_{\mathrm{p}, \text { hot }}= & {\left[2 \mathrm{~L}_1 \mathrm{~L}_2 \mathrm{~N}_{\mathrm{p}}2 \delta \mathrm{L}_2 \mathrm{n}_{\mathrm{f} 1}\right]+\left[2 \mathrm{~b}_1 \mathrm{~L}_2 \mathrm{~N}_{\mathrm{p}}\right]+\left[2\left(\mathrm{~b}_2+\right.\right.} \\ & \left.\left.2 \delta_{\mathrm{w}}\right) \mathrm{L}_1\left(\mathrm{~N}_{\mathrm{p}}+1\right)\right] \mathrm{A}_{\mathrm{p}, \text { hot }}=529.135 \mathrm{~m}^2\end{aligned}$
·Calculate the number of offset strip fins for hot fluid (per the number of fins) by: $\mathrm{n}_{\mathrm{off}, \text { hot }}=\frac{L_2}{\lambda_1}=500$
·The total fin area A_{f1} for hot fluid is obtained using Equation:
$\begin{gathered}\mathrm{A}_{\mathrm{f}, \text { hot }}=2\left(\mathrm{~b}_1\delta\right) \mathrm{L}_2 \mathrm{n}_{\mathrm{f} 1}+2\left(\mathrm{~b}_1\delta\right) \delta \mathrm{n}_{\mathrm{off} 1} \mathrm{n}_{\mathrm{f} 1}+ \\ \left(\mathrm{p}_{\mathrm{f} 1}\delta\right) \delta\left(\mathrm{n}_{\mathrm{off} 1}1\right) \mathrm{n}_{\mathrm{f} 1}+2 \mathrm{p}_{\mathrm{f} 1} \delta \mathrm{n}_{\mathrm{f} 1} \\ \mathrm{~A}_{\mathrm{f}, \text { hot }}=2015.02 \mathrm{~m}^2 \\ \mathrm{~A}_{\text {total,hot side }}=\mathrm{A}_{\mathrm{p}, \text { hot }}+\mathrm{A}_{\mathrm{f}, \text { hot }}=2544.159 \mathrm{~m}^2\end{gathered}$
This is larger required by: $\frac{2544.1592447.648}{2447.648}=3.94 \%$ (This percentage is acceptable to compensate any loss of efficiency of heat transfer). Other geometrical characteristics of interest are:
The freeflow area on the air side is calculated by subtracting the frontal area on the air side from the area blocked by the fins at the core's entry on that side.
$\mathrm{A}_{\mathrm{c}, \text { hot }}=\left(\mathrm{b}_1\delta\right) *\left(\mathrm{p}_{\mathrm{f}}\delta\right) * \mathrm{n}_{\mathrm{f}}=0.37 \mathrm{~m}^2$
·The frontal area for hot stream: $\mathrm{A}_{\mathrm{fr}}=\mathrm{L}_1 * \mathrm{~L}_3=1.375 \mathrm{~m}^2$
·The porosity $(\sigma)$: $\sigma=\frac{\mathrm{A}_{\mathrm{c}}}{\mathrm{A}_{\mathrm{fr}}}=0.27$
·The surface area density (β) is: $\beta=\frac{A_{\text {hot total }}}{\mathrm{b}_1 \mathrm{~N}_{\mathrm{b}} \mathrm{L}_1 \mathrm{~L}_2}=1392 \frac{\mathrm{m}^2}{\mathrm{~m}^3}$
·The hydraulic diameter (D_{h}) is: $\mathrm{D}_{\mathrm{h}, \text { hot side }}=\frac{4 * \mathrm{~L}_2 * \mathrm{~A}_{\mathrm{c}}}{\mathrm{A}_{\text {hot total }}}=0.0023 \mathrm{~m}$
3.5.2 Calculations of cold side
·The number of passages for the cold side=N_{p}+1=64
·The total number of fins for cold side are calculated by: $\mathrm{n}_{\mathrm{f} 2}=\frac{\mathrm{L}_1}{\mathrm{p}_{\mathrm{f}}}\left(\mathrm{N}_{\mathrm{p}}+1\right)=51.599 * 10^3$
·The total primary area (A_{p})for cold side calculated by: $\mathrm{A}_{\mathrm{p}, \text { cold }}=\left[2 \mathrm{~L}_1 \mathrm{~L}_2\left(\mathrm{~N}_{\mathrm{p}}+1\right)2 \delta \mathrm{L}_2 \mathrm{n}_{\mathrm{f} 2}\right]+\left[2 \mathrm { b } _ { 2 } \mathrm { L } _ { 2 } \left(\mathrm{N}_{\mathrm{p}}+\right.\right.1) ]+\left[2\left(\mathrm{~b}_1+2 \delta_{\mathrm{w}}\right) \mathrm{L}_1 \mathrm{~N}_{\mathrm{p}}\right]$
$A_{\mathrm{p}, \mathrm{cold}}=519.12 \mathrm{~m}^2$
·Calculate the number of offset strip fins: $\mathrm{n}_{\mathrm{off}, \mathrm{cold}}=\frac{\mathrm{L}_2}{\lambda_2}=575$
·The total fin area A_{f1} for hot fluid is obtained using Equation:
$\begin{gathered}\mathrm{A}_{\mathrm{f}, \text { cold }}=2\left(\mathrm{~b}_2\delta\right) \mathrm{L}_2 \mathrm{n}_{\mathrm{f} 2}+2\left(\mathrm{~b}_2\delta\right) \delta \mathrm{n}_{\mathrm{off} 2} \mathrm{n}_{\mathrm{f} 2}+ \\ \left(\mathrm{p}_{\mathrm{f} 2}\delta\right) \delta\left(\mathrm{n}_{\mathrm{off} 2}1\right) \mathrm{n}_{\mathrm{f} 2}+2 \mathrm{p}_{\mathrm{f} 2} \delta \mathrm{n}_{\mathrm{f} 2} \\ \mathrm{~A}_{\mathrm{f}, \mathrm{cold}}=2053.58 \mathrm{~m}^2 \\ \mathrm{~A}_{\mathrm{t}}=\mathrm{A}_{\mathrm{p}, \mathrm{cold}}+\mathrm{A}_{\mathrm{f}, \mathrm{cold}} \\ \mathrm{A}_{\text {total,cold side }}=519.12+2053.58=2574.26 \mathrm{~m}^2\end{gathered}$
This is larger required by: $\frac{2574.262447.648}{2447.648}=5 \%$, and other geometrical characteristics of interest are:
·The freeflow area on the cold side: $\mathrm{A}_{\mathrm{c}, \mathrm{cold}}=\left(\mathrm{b}_2\delta\right) *\left(\mathrm{p}_{\mathrm{f} 2}\delta\right) * \mathrm{n}_{\mathrm{f} 2}=0.3 \mathrm{~m}^2$
·The frontal area for cold streams: A_{fr}=L_{1}*L_{3}=1.375 m^{2}
·The porosity for cold streams: $\sigma=\frac{A_c}{A_{\mathrm{fr}}}=0.218$
·The surface area density for cold streams:
$\beta=\frac{\mathrm{A}_{\text {cold total }}}{\mathrm{b}_2 \mathrm{~N}_{\mathrm{b}} \mathrm{L}_1 \mathrm{~L}_2}=\frac{2574.26}{5 * 10^{3} * 63 * 1.2 * 4.03}=1689 \frac{\mathrm{m}^2}{\mathrm{~m}^3}$
The hydraulic diameter (D_{h}), $\mathrm{D}_{\mathrm{h}, \text { cold side }}=\frac{4 * \mathrm{~L}_2 * \mathrm{~A}_{\mathrm{c}}}{\mathrm{A}_{\text {cold total }}}=0.00188 \mathrm{~m}$.
3.5.3 Calculations of pressure drops
It can estimate pressure drop for hot side (air stream):
·Calculate the volumetric flow rate of air, $\dot{\mathrm{V}}_{\mathrm{air}}=\frac{\dot{\mathrm{m}}_{\mathrm{air}}}{\rho_{\mathrm{air}}}=0.485 \mathrm{~m}^3 / \mathrm{s}$
·Calculate the velocity of air, $\mathrm{u}_{\mathrm{air}}=\frac{\mathrm{V}_{\mathrm{air}}}{\mathrm{A}_{\mathrm{c} 1}}=1.31 \frac{\mathrm{m}}{\mathrm{s}}$
·Calculate the Reynolds number, $\operatorname{Re}=\frac{\rho \mathrm{uD}_{\mathrm{h}}}{\mu}=1805.97$
·Calculate the friction factor $\mathrm{f}_{\text {hot }}($ For $120 \leqslant \operatorname{Re} \leqslant 104)$ on the hot side (air) is:
$\mathrm{f}_1=9.6243 \mathrm{Re}^{0.7422}\left(\frac{\mathrm{p}_{\mathrm{f}}\delta}{\mathrm{b}\delta}\right)^{0.1856}\left(\frac{\delta}{\lambda}\right)^{0.3053}\left(\frac{\delta}{\mathrm{p}_{\mathrm{f}}\delta}\right)^{0.2659}$$\begin{gathered}{\left[1+7.669 * 10^{8} \operatorname{Re}^{4.429}\left(\frac{\mathrm{p}_{\mathrm{f}}\delta}{\mathrm{b}\delta}\right)^{0.92}\left(\frac{\delta}{\lambda}\right)^{3.767}\left(\frac{\delta}{\mathrm{p}_{\mathrm{f}}\delta}\right)^{0.236}\right]^{0.1}} \\ {\left[\text { Get: }\left(\frac{\mathrm{p}_{\mathrm{f}}\delta}{\mathrm{b}\delta}\right)=0.27,\left(\frac{\delta}{\lambda}\right)=0.031,\left(\frac{\delta}{\mathrm{p}_{\mathrm{f}}\delta}\right)=0.16\right]} \\ \therefore \mathrm{f}_1=0.033\end{gathered}$
·Calculate the mass velocity $\left(G_{\text {air }}\right)$ on the hot side, $\mathrm{G}_{\text {air }}=\frac{\dot{\mathrm{m}}_{\mathrm{air}}}{\mathrm{A}_{\mathrm{c}}}=12.95 \frac{\mathrm{kg}}{\mathrm{m}^2 \mathrm{~s}}, \rho_{\text {in }}=7.573762\left(\mathrm{~kg} / \mathrm{m}^3\right), \rho_{\text {out }}=9.886560384\left(\mathrm{~kg} / \mathrm{m}^3\right)$
·Calculate the ($\left(\rho_{m}\right),\left(\frac{1}{\rho_{\mathrm{m}}}\right)=\frac{1}{2} *\left(\frac{1}{\rho_{\text {in }}}+\frac{1}{\rho_{\text {out }}}\right), \rho_{\mathrm{m}}=8.576\left(\mathrm{~kg} / \mathrm{m}^3\right)$
$\begin{aligned} & \mathrm{K}_{\mathrm{in}}=0.40461538+0.00817016 \sigma0.41317016 \sigma^2 =0.376 \\ & \mathrm{~K}_{\mathrm{out}}=0.998951051.96088578 \sigma+0.96270396 \sigma^2 =0.539 \\ & \Delta \mathrm{P}=\frac{\mathrm{G}^2}{2 * \rho_{\text {in }}} *\left[\left(1\sigma^2+\mathrm{K}_{\mathrm{in}}\right)+2\left(\frac{\rho_{\text {in }}}{\rho_{\text {out }}}1\right)+\frac{4 \mathrm{fL}_2}{\mathrm{D}_{\mathrm{h}}}\right. \\ & \left.*\left(\frac{\rho_{\mathrm{in}}}{\rho_{\mathrm{m}}}\right)\left(1\sigma^2\mathrm{K}_{\text {out }}\right) *\left(\frac{\rho_{\text {in }}}{\rho_{\text {out }}}\right)\right] \\ & \therefore \Delta \mathrm{P}_{\mathrm{air}}=2267 \mathrm{~Pa} \\ & \end{aligned}$
To estimate pressure drop for cold side, with same way calculated the pressure drop for cold side streams as shown in Table 10.
Table 10. Pressure drop for cold side streams
Streams 
Velocity (m⁄s) 
Re 
∆P (Pa) 
LPN_{2} 
0.943 
197.85 
1487 
WN_{2} 
6.06 
1208.6 
11145 
GO_{2} 
1.46 
332.2 
1944.96 
HPN_{2} 
0.118 
108.73 
101 
4.1 Rating of design (The efficiency)
Based on the above calculations, it can calculate the efficiency of the system under consideration:
Using data of air stream, where Cp_{air}=1003.87, Pr_{air}=0.64, and when 400≤Re≤5500, $\begin{aligned} & \mathrm{j}=2.11497 * 10^{2}1.02089 * 10^{5} \mathrm{Re}+2.37311 * 10^{9} \mathrm{Re}^21.89734 * 10^{13} \mathrm{Re}^3, \mathrm{j}=0.00933\end{aligned}$.
·Calculate the heat transfer coefficient $(\alpha): \alpha=\mathrm{jGc}_{\mathrm{p}} \operatorname{Pr}^{2 / 3}=163.32 \frac{\mathrm{w}}{\mathrm{m}^2 \cdot \mathrm{k}}$
where, k_{f} is thermal conductivity, $\mathrm{k}_{\mathrm{f}}=88: 251 \frac{\mathrm{w}}{\mathrm{m} \cdot \mathrm{K}}$ , for all alloys [73]. k_{f }Average)=169.5$\frac{w}{m \cdot K}$
·Calculate fin parameter (m), $\mathrm{m}=\sqrt{\frac{2 \alpha}{\mathrm{k}_{\mathrm{f}} \delta}\left(1+\frac{\delta}{\lambda}\right)}=89.158$
·Calculate the fin profile length, $\mathrm{L}_{\mathrm{f}}=\frac{\mathrm{b}_1}{2}\delta=0.00275 \mathrm{~m}$
·Calculate the fin efficiency (η_{f}), $\eta_{\mathrm{f}}=\frac{\tanh \left(\mathrm{mL}_{\mathrm{f}}\right)}{\mathrm{mL}_{\mathrm{f}}}=98 \%$
The overall surface efficiency $\left(\eta_{\text {overal HT }}\right)=1\frac{\mathrm{A}_{\mathrm{f}}}{\mathrm{A}_{\mathrm{HT}}}\left(1\eta_{\mathrm{f}}\right)=98.4 \%$.
4.2 MATLAB code
A computer code has been written based on the steps summed up in the preceding sections. The program has been coded using the MATLAB language and has a mainline and seven subroutines. A flowchart of code work (Figure 10) shows a platefin multistream heat exchanger design. The program calculates the optimum required area of heat transfer, fin geometry, and overall exchanger sizing. The advantage of using code lies in its capability for handling every one of the streams in the exchanger in an individual manner and providing the profiles of the pressure, temperature, heat transfer coefficients, Reynolds number, and so on, at the stage of the sizing. The code included sizing and rating processes.
Figure 10. Flow chart of the code work
4.3 Validation the results by aspen EDR software
The validation of the data gathered in the case study to design a heat exchanger is presented in this part. The outcomes of the chosen design technique are evaluated against those obtained using advanced simulation tools. The rigorous simulation software for comparison was Aspen Exchanger Design and Rating (Aspen EDR) Version 10.0. EDR provides detailed simulations of multistream platefin heat exchangers based on thermalhydraulic correlations.
The findings obtained using data from Boehme et al. [14] have been validated and compared with those obtained using the rigorous simulation tool EDR. In this case study, the number of streams is five. One stream is hot and the others are cold. At the start of the EDR menu, you can choose the computation mode as well as the number of streams and project title, as shown in Figure 11.
Figure 11. The selecting of the calculation mode and the details of number of streams in EDR software
The data entry process for the streams in case one is shown in Figure 12.
Figure 12. The data entry process of mass flow rate, temperatures, and inlet pressures of the streams in EDR software
After assembling the input data, it is executed with the display of errors and warnings (if any), and the results may be seen in a summary or in detail. Figures 13 and 14 are shown as follows.
Figure 13. The full results1 in detail
Figure 14. The full results2 in detail
4.4 Comparison with previous results
In this section, comparison of data from the current study and a previous study for a heat exchanger by Boehme et al. [14]. Table 11 displays the comparison results.
Table 11. Comparison of data from the current study and a previous study for a heat exchanger by Boehme et al. [14]
Stream 
Air 
WN_{2} 
GO_{2} 
LPN_{2} 
Hot end temperature (K) 
313.05 313 
307.76 306 
307.19 306 
307.14 306.07 
Previous study 

Present design 

Cold end temperature (K) 
198.48 198.98 
193.15 191 
193.15 191 
193.15 191 
Previous study 

Present design 

Heat transfer area (m^{2}) 
1453.98 2447.64 
1453.9 1596.3 
485.34 433.9 
259.32 262.11 
Previous study 

Present design 

Rate of heat transfer (kW) 
555.21 555.616 
363.97 362.38 
98.67 98.50 
59.43 59.50 
Previous study 

Present design 

Core pressure drop (Pa) 
3861 2264 
10333 6819 
5581 2183 
10220 10680 
Previous study 

Present design 
Although the design of the heat exchanger has been covered in earlier research, there is still a need for many more studies to find the best design. This is the motivation for the current study. The platefin heat exchanger (PFHE), which is a crucial part of the design of air separation units, is the subject of the current study. In order to reduce energy consumption, the air entering this operation has been cooled using energy from streams coming from the distillation tower in the air separation unit (ASU). Based on one of the models, a multistream platefin heat exchanger (MSPFHE) was created. In the beginning, MATLAB code was created utilizing a certain model's technique, and it was then applied to several realworld scenarios. This code has been successfully used to size platefin heat exchangers in a range of situations, including calculations of thermodynamic property, heat transfer, pressure drop, fintype selection, and final heat exchanger dimensions. In this case as well, inconsistency has been chosen as a key component of achieving the high efficacy that this kind demands. The purpose of the project is to develop and create a multistream platefin heat exchanger (MSPFHE) that may be used in the cold box of an air separation unit practically and without limitations. Finally, the effectiveness of the Aspen EDR software in simulating the same environmental circumstances that nature produces has been evaluated and the results have been validated. The main dimensions of the heat exchanger and details of the rest of the work were determined using an analytical solution and code.
The heat exchanger was designed, and the main dimensions of the exchanger, as well as the rest of the work details, were obtained using an analytical solution and code. Based on the current findings, the efficiency was calculated and found to be higher than 98%, which is required in this type of application, as well as a validation of the results by comparing them with previous literature. On the other hand, an investigation using the EDR program was conducted. Finally, the results of this work were compared to those of previously published studies, and the validity of the proposed program showed that the results were very acceptable.
A 
Area (m^{2}) 
A_{f} 
Total fin area (m^{2}) 
A_{p} 
Total primary area (m^{2}) 
A_{t} 
Total area (m^{2}) 
b 
Fin height (m) 
c_{p} 
Specific heat ( $\left(\mathrm{Jkg}^{1} \mathrm{~K}^{1}\right)$ ) 
C 
Heat capacity ($\mathrm{WK}^{1}$) 
D_{h} 
Hydraulic diameter (m) 
f 
Fanning friction factor 
j 
Colburn factor 
Re 
Reynolds number 
G 
Mass flow velocity ($\mathrm{kgs}^{1} \mathrm{~m}^{2}$) 
L_{f} 
Fin profile length (m) 
L_{1} 
Length of the core HE (m) 
δ_{w} 
plate thickness (m) 
μ 
Dynamic viscosity coefficient ( $\mathrm{Nsm}^{2}$) 
ρ 
Density ($\mathrm{kgm}^{3}$) 
η_{f} 
Fin efficiency 
L_{2} 
Width of the core of HE (m) 
L_{3} 
Core height(m) 
m 
Fin parameter ($m^{1}$) 
$\dot{m}$ 
Mass flow rate ($\mathrm{kgs}^{1}$) 
n_{f} 
Number of fins 
n_{off} 
Number of offset fins 
∆P 
Pressure drop (Pa) 
N_{p} 
Number of passages 
N_{f} 
Fin densities (Fins/m) 
P 
Pressure (Pa) 
Pr 
Prandtl number 
U 
Overall heat transfer Coefficient ($\mathrm{Wm}^{2} \mathrm{~K}^{1}$) 
T 
Temperature (K) 
u 
Velocity $\left(\mathrm{m}\, \mathrm{s}^{1}\right)$ 
p_{f} 
Fin pitch (m) 
α 
Heat transfer coefficient $\left(\mathrm{Wm}^{2} \mathrm{~K}^{1}\right)$ 
δ 
Fin thickness (m) 
η_{overal HT} 
Overall surface fin efficiency 
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